Chain complex

A (co) chain complex in mathematics is a sequence of Abelian groups or modules or - even more generally - objects in an Abelian category , which are linked in a chain-like manner by illustrations . ${\displaystyle R}$

definition

Chain complex

A chain complex consists of a sequence

${\displaystyle \,C_{n},\,n\in \mathbb {Z} }$

of modules (Abelian groups, objects of an Abelian category A ) and a sequence ${\displaystyle R}$

${\displaystyle d_{n}:C_{n}\rightarrow C_{n-1}}$

of module homomorphisms (group homomorphisms, morphisms in A ), so that ${\displaystyle R}$

${\displaystyle \,d_{n}\circ d_{n+1}=0}$

holds for all n . The operator is called the boundary operator . Elements of hot n-chains . Elements of ${\displaystyle \mathrm {d} _{n}}$ ${\displaystyle C_{n}}$

${\displaystyle Z_{n}(C,d):=\ker d_{n}\subseteq C_{n}}$ or. ${\displaystyle B_{n}(C,d):=\mathop {\mathrm {im} } d_{n+1}\subseteq C_{n}}$

are called n-cycles or n-boundaries . Because of the condition , each edge is a cycle. The quotient ${\displaystyle \,d_{n}d_{n+1}=0}$

${\displaystyle \,H_{n}(C,d):=Z_{n}(C,d)/B_{n}(C,d)}$

is called the n th homology group (homology object) of , its elements are called homology classes . Cycles that are in the same homology class are called homologous . ${\displaystyle \,(C,d)}$

Coquette complex

A coquette complex consists of a sequence

${\displaystyle C^{n},\,n\in \mathbb {Z} }$

of modules (Abelian groups, objects of an Abelian category A ) and a sequence ${\displaystyle R}$

${\displaystyle \,d^{n}:C^{n}\rightarrow C^{n+1}}$

of module homomorphisms (group homomorphisms, morphisms in A ) so that ${\displaystyle R}$

${\displaystyle \,d^{n}\circ d^{n-1}=0}$

holds for all n . Elements of hot n-coquettes . Elements of ${\displaystyle \,C^{n}}$

${\displaystyle \,Z^{n}:=\ker d^{n}\subseteq C^{n}}$ or. ${\displaystyle B^{n}:=\operatorname {im} d^{n-1}\subseteq C^{n}}$

are called n-coccycles or n-corands . Because of the condition , every Korand is a cocycle. The quotient ${\displaystyle \,d^{n}d^{n-1}=0}$

${\displaystyle \,H^{n}(C,d):=Z^{n}(C,d)/B^{n}(C,d)}$

is called the n th cohomology group (cohomology object) of , its elements cohomology classes . Coccycles that are in the same cohomology class are called cohomologous . ${\displaystyle \,(C,d)}$

Double complex

A double complex

A double-complex     in the abelian category A is essentially a chain complex in the abelian category of chain complexes in A . Somewhat more precisely consists     of objects ${\displaystyle D_{**}}$ ${\displaystyle D_{**}}$

${\displaystyle D_{p,q}\in \operatorname {ob} A\,,\quad p,q\in \mathbb {Z} }$

along with morphisms

${\displaystyle D_{p,q}{\xrightarrow {d}}D_{p-1,q}}$   and    ${\displaystyle D_{p,q}{\xrightarrow {d'}}D_{p,q-1}\quad \forall \,p,q\in \mathbb {Z} }$

which meet the following three conditions:

${\displaystyle d\circ d=0\quad d'\circ d'=0\quad d\circ d'+d'\circ d=0\,.}$

The total complex of     the double     complex is given by the chain complex ${\displaystyle \operatorname {Tot} (D)_{*}}$ ${\displaystyle D_{**}}$

${\displaystyle \operatorname {Tot} (D)_{n}=\bigoplus _{p+q=n}D_{p,q}}$

with the following border illustration: for     with     is ${\displaystyle x\in D_{p,q}}$ ${\displaystyle p+q=n}$

${\displaystyle d_{n}(x)=d(x)+d'(x)\in D_{p-1,q}\oplus D_{p,q-1}\subseteq \operatorname {Tot} (D)_{n-1}\,.}$

Double complexes are needed, among other things, in order to prove that the value of is independent of whether one M dissolves or N . ${\displaystyle \operatorname {Tor} _{*}^{R}(M,N)}$

properties

• A chain complex is exactly at the point if is, correspondingly for coquette complexes. The (co) homology thus measures how much a (co) chain complex deviates from the exactness. ${\displaystyle (C_{\bullet },d_{\bullet })}$ ${\displaystyle i}$ ${\displaystyle H_{i}(C_{\bullet },d_{\bullet })=0}$

• A chain complex is called acyclic if all of its homology groups disappear, i.e. if it is exact.

Chain homomorphism

One function

${\displaystyle f:(A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet })}$

is called (Ko) -chain homomorphism , or simply chain mapping , if it consists of a sequence of group homomorphisms , which is exchanged with the boundary operator . That means for the chain homomorphism: ${\displaystyle f_{n}:A_{n}\rightarrow B_{n}}$ ${\displaystyle d}$

${\displaystyle d_{B,n}\circ f_{n}=f_{n-1}\circ d_{A,n}}$.

The same applies to the coquette homomorphism

${\displaystyle d_{B}^{n}\circ f_{n}=f_{n+1}\circ d_{A}^{n}}$.

This condition ensures that cycle maps to cycle and edges to edges. ${\displaystyle f}$

Chain complexes together with the chain homomorphisms form the category Ch (MOD R) of the chain complexes.

Euler characteristic

It is a coquette complex of modules over a ring . If only finitely many cohomology groups are nontrivial, and if these are finite-dimensional, then the Euler characteristic of the complex is defined as the integer ${\displaystyle (C,d)}$ ${\displaystyle R}$ ${\displaystyle R}$

${\displaystyle \chi (C,d)=\sum _{i}(-1)^{i}\dim _{K}\mathrm {H} ^{i}(C,d)\in \mathbb {Z} .}$

If the individual components are also finite-dimensional and only finitely many of them are nontrivial, so is ${\displaystyle C^{i}}$

${\displaystyle \chi (C,d)=\sum _{i}(-1)^{i}\dim _{K}C^{i}\in \mathbb {Z} .}$

In the special case of a complex with only two nontrivial entries, this statement is the ranking . ${\displaystyle C^{0}\to C^{1}}$

In a somewhat more general way, a chain complex is called perfect if only finitely many components are nontrivial and each component is a finitely generated projective module . The dimension is then to be replaced by the associated equivalence class in the K ₀ group of and it is defined as the Euler characteristic ${\displaystyle C^{i}}$ ${\displaystyle R}$

${\displaystyle \chi (C,d)=\sum _{i}(-1)^{i}[C^{i}]\in K_{0}(R).}$

If every projective module is free , for example if there is a body or a main ideal ring , then one can speak of dimensions and receive with . Then this more general definition coincides with the one given first. ${\displaystyle R}$ ${\displaystyle K_{0}(R)\cong \mathbb {Z} }$ ${\displaystyle [R^{n}]{\mathrel {\hat {=}}}n}$

Examples

• Simplicial complex
• The singular chain complex for the definition of the singular homology and the singular cohomology of topological spaces .
• Group (co) homology .
• Each homomorphism defines a coquette complex ${\displaystyle f\colon A\to B}$

${\displaystyle (C,d)=(\ldots \to 0\to 0\to A\to B\to 0\to 0\to \ldots ).}$

If one sets the indices so that it is in degree 0 and in degree 1, then is ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle H^{0}(C,d)=\ker f}$ and ${\displaystyle H^{1}(C,d)=\mathrm {coker} \,f.}$

The Euler characteristic

${\displaystyle \dim \ker f-\dim \mathrm {coker} \,f}$

of is in the theory of Fredholm operators of Fredholm index of called. This denotes the coke core of . ${\displaystyle (C,d)}$ ${\displaystyle f}$ ${\displaystyle \mathrm {coker} \,f}$ ${\displaystyle f}$

• An elliptical complex or a Dirac complex is a coquette complex that is important in global analysis . These occur, for example, in connection with the Atiyah-Bott fixed point theorem .

literature

• Peter John Hilton, Urs Stammbach : A Course in Homological Algebra ( Graduate Texts in Mathematics 4). Springer, New York a. a. 1971, ISBN 0-387-90033-0 .

Individual evidence

1. ↑ P. 7–8 in Charles A. Weibel : An introduction to homological algebra (=  Cambridge Studies in Advanced Mathematics . Volume 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 .
2. ↑ Section 2.7 in Charles A. Weibel : An introduction to homological algebra (=  Cambridge Studies in Advanced Mathematics . Volume 38 ). Cambridge University Press, 1994, ISBN 0-521-43500-5 .
3. ↑ J. Cuntz, R. Meyer, J. Rosenberg: Topological and Bivariant K-Theory , Birkhäuser Verlag (2007), ISBN 3-764-38398-4 , definition 1.31